Coulomb Wave Functions Research Articles

Total cross sections in five methods for two-electron capture by alpha particles from helium: CDW-4B, BDW-4B, BCIS-4B, CDW-EIS-4B and CB1-4B

This work is on quantum-mechanical four-body distorted wave theories for double electron capture in collisions between fast heavy multiply charged ions and heliumlike atomic systems. The five widely used distorted wave methods of the first- and second-order in the perturbation series expansions are compared with the available experimental data on alpha –He collisions. These are the four-body boundary-corrected first Born (CB1-4B), the boundary-corrected continuum intermediate state (BCIS-4B), the Born distorted wave (BDW-4B), the continuum distorted wave (CDW-4B) and the continuum distorted wave-eikonal initial state (CDW-EIS-4B) methods.We address the complete breakdown of the CDW-EIS-4B method at all impact energies within its expected validity domain (100–10000 keV). Further, the relative performance is evaluated of the second-order theories with and without the eikonalization of the two-electron Coulomb wavefunctions for double continuum intermediate states. Finally, at all the considered intermediate and high energies, the practical aspects of the studied five methods are investigated by protracted evaluations of the convergence rates of total cross sections as a function of the number of quadrature points per axis in numerical computations of multi-dimensional (3D-5D) integrals.

A generating integral for the matrix elements of the Coulomb Green’s function with the Coulomb wave functions

We analytically evaluate the generating integral Knl(β,β′)=∫0∞∫0∞e−βr−β′r′Gnl(r,r′)rqr′q′drdr′ and integral moments Jnl(β,r)=∫0∞dr′Gnl(r,r′)r′qe−βr′ for the reduced Coulomb Green’s function Gnl(r, r′) for all values of the parameters q and q′, when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions.

Complementary Romanovski–Routh Polynomials, Orthogonal Polynomials on the Unit Circle, and Extended Coulomb Wave Functions

In a recent paper (Martinez-Finkelshtein et al. in Proc Am Math Soc 147:2625–2640, 2019) some interesting results were obtained concerning complementary Romanovski–Routh polynomials, a class of orthogonal polynomials on the unit circle and extended regular Coulomb wave functions. The class of orthogonal polynomials here are generalization of the class of circular Jacobi polynomials. In the present paper, in addition to looking at some further properties of the complementary Romanovski–Routh polynomials and associated orthogonal polynomials on the unit circle, behaviour of the zeros of these extended Coulomb wave functions are also studied.

Some integrals involving Coulomb functions associated with the three-dimensional proper Lorentz group

For two continual bases in the representation space, we obtain the matrix elements of the linear operator transforming the first basis into the second. These elements are expressed in terms of Coulomb wave functions. Computing the matrix elements of subrepresentations to some subgroups or their separate elements and using the connection between above bases, we evaluate some integrals involving Coulomb wave functions.

Quasirelativistic Coulomb Sturmian basis sets for atomic physics problems: illustration of their applicability to two-photon decay rates in hydrogen-like atoms

The continuum and discrete quasirelativistic one-electron Sturmian basis sets associated with the second-order Dirac–Coulomb equation are constructed by solving appropriate Sturm–Liouville problems. The potential-weighted orthogonality and closure relations obeyed by the members of such sets are discussed. The special case where both quasirelativistic Coulomb Sturmian basis sets are used to obtain solutions to the Gell-Mann–Feynman equation is presented. It is proved that for the continuum quasirelativistic Sturm–Liouville problem, the spectrum of potential strengths is continuous and covers the whole real axis. We have also introduced the complete outgoing continuum quasirelativistic Coulomb wavefunctions and presented the discretization of its radial components. The compact representation of the fully relativistic and quasirelativistic Coulomb Green’s functions is outlined. By making use of the Sturmian expansion of the Coulomb Green’s functions, closed-form expressions of two-photon spontaneous decay rates are derived for arbitrary multipole channels. For the sake of assessing the domain of applicability of quasirelativistic basis sets, an illustrative application is then carried out for two-photon decay rates for hydrogen-like ions. A comparison of present results with reference values previously obtained within a fully relativistic scheme (Bona et al 2014) is undertaken.

Balmer emission induced by proton impact on atomic hydrogen

The semiclassical two-centre convergent-close-coupling approach is applied to study Balmer emission in proton-hydrogen scattering at the incident proton energies from 5–1 MeV. The approach uses wave-packet pseudostates for the discretization of the continuous spectrum of the hydrogen atom, constructed from the Coulomb wave function. All cross sections for target excitation into the final states with principal quantum numbers n = 3 and 4 required for obtaining the Balmer emission cross sections, polarisation fraction and Balmer decrement are calculated. Corresponding electron-capture cross sections are also given. A substantial variation in the cross sections for population of magnetic sublevels obtained in different theoretical approaches is found. The present cross section for excitation of the n = 3 shell as a whole does not agree with experiment, but supports earlier calculations. At the same time, the individual cross section for excitation of the 3p state displays excellent agreement with available experimental measurements. The results for polarisation fraction of the Balmer-α emission significantly disagree with experimental measurements at high energies. The calculated Balmer decrement plateaus at about 100 keV and can be used in astrophysical applications.

Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions

We consider properties and applications of a sequence of polynomials known as complementary Romanovski-Routh polynomials (CRR polynomials for short). These polynomials, which follow from the Romanovski-Routh polynomials or complexified Jacobi polynomials, are known to be useful objects in the studies of the one-dimensional Schr\"{o}dinger equation and also the wave functions of quarks. One of the main results of this paper is to show how the CRR-polynomials are related to a special class of orthogonal polynomials on the unit circle. As another main result, we have established their connection to a class of functions which are related to a subfamily of Whittaker functions that includes those associated with the Bessel functions and the regular Coulomb wave functions. An electrostatic interpretation for the zeros of CRR-polynomials is also considered.

Hedge Fund Investing or Mutual Fund Investing: An Application of Multi-Attribute Utility Theory

This paper contrasts high-risk, hedge fund trading, with low-risk, mutual fund trading, in terms of their differing utility functions. We envision hedge funds, led by informed traders who use information to seek out investment opportunities, timing market conditions, with the expectation that prices will move in their favor. Directional hedge funds act to influence prices, while non-directional hedge funds do not act to influence prices. We present utility functions based on steeply-sloping Laplace distributions and hyperbolic cosine distributions, to describe the actions of directional hedge fund traders. Less steeply-sloping lognormal distributions, Coulomb wave functions, quadratic utility functions, and Bessel utility functions are used to describe the investing style of non-directional hedge fund traders. Flatter Legendre utility functions and inverse sine utility functions describe the modest profit-making aspirations of mutual fund traders. The paper’s chief contribution is to develop optimal prices quantitatively, by intersecting utility functions with price distributions. Price distributions for directional hedge fund returns are portrayed as sharp increases and decreases, in the form of jumps, in a discrete arrival Poisson-distributed process. Separate equations are developed for directional hedge fund strategies, including event-driven arbitrage, and global macro strategies. Non-directional strategies include commodity trading, risk-neutral arbitrage, and convertible arbitrage, with primarily lognormal pricing distributions, and some Poisson jumps. Mutual funds are perceived to be Markowitz portfolios, lying on the Capital Market Line, or the International Capital Market Line, tangent to the Efficient Frontier of minimum variance-maximum return portfolios.

An Improved Analysis of Masses and Decay Constants of Heavy Flavour Mesons within Variational Approach

We employ the variational method to study the properties such as masses, decay constants, oscillation frequency and branching ratios of leptonic decays of heavy flavour mesons with linear cum coulomb Cornell potential. Gaussian function, Coulomb wave function and Airy function are taken as the trial wave-function of variational method in this study. Our analysis suggests that Gaussian trial wave-function provides results which are in close proximity with the experimental results. We also make a comparison with the results from QCD Sum rules and lattice QCD, as well as with recent PDG data.

Imaginary Time Density Functional Calculation of Ground States of Atoms Using CWDVR Approach

The self-consistent Kohn-Sham equations for many-electron atoms are solved using the Coulomb wave function Discrete Variable Method (CWDVR). Wigner type functional is used to incorporate correlation functional. The discrete variable method is used for the uniform and optimal spatial grid discretization and solution of the Kohn-Sham equation. The equation is numerically solved using the CWDVR method. First time we have reported the solution of the Kohn-Sham equation on the ground state problem for the many-electronic atoms by the CWDVR method. Our results suggest CWDVR approach shown to be an efficient and precise solution of ground-state energies of atoms. We illustrate that the calculated electronic energies for He, Li, Be, B, C, N and O atoms are in good agreement with other best available values.

Connection formulas between Coulomb wave functions

The mathematical relations between the regular Coulomb function Fηℓ(ρ) and the irregular Coulomb functions Hηℓ±(ρ) and Gηℓ(ρ) are obtained in the complex plane of the variables η and ρ for integer or half-integer values of ℓ. These relations, referred to as “connection formulas,” form the basis of the theory of Coulomb wave functions and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function Fηℓ(ρ) are studied, in particular, under the transformation ℓ ↦ −ℓ − 1, by means of the modified Coulomb function Φηℓ(ρ), which is entire in the dimensionless energy η−2 and the angular momentum ℓ. Then, it is shown that, for integer or half-integer ℓ, the irregular functions Hηℓ±(ρ) and Gηℓ(ρ) can be expressed in terms of the derivatives of Φη,ℓ(ρ) and Φη,−ℓ−1(ρ) with respect to ℓ. As a consequence, the connection formulas directly lead to the description of the singular structures of Hηℓ±(ρ) and Gηℓ(ρ) at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of η−1.

Double ionization of water molecules by proton impact: the role of the direct ionization mechanism

The proton-induced double ionization process of water molecules is here theoretically investigated at high impact energy. The current 1st Born perturbative approach refers to an initial collisional state that both includes a target molecular wave function expressed as a single-center linear combination of atomic orbitals and a plane wave for describing the incident proton whereas the final collisional state is characterized by a scattered proton modeled by a plane wave and two ejected electrons modeled by Coulomb wave functions coupled via a Gamow factor. Total and partial cross sections are then provided and compared with existing experimental and theoretical data in considering the various dissociative channels. Additionally, a study of the energy dependence of the double ionization process is reported for impact energies ranging from 300 keV to 10 MeV.

Effective-range function methods for charged particle collisions

Different versions of the effective-range function method for charged particle collisions are studied and compared. In addition, a novel derivation of the standard effective-range function is presented from the analysis of Coulomb wave functions in the complex plane of the energy. The recently proposed effective-range function denoted as $\Delta_\ell$ [Phys. Rev. C 96, 034601 (2017)] and an earlier variant [Hamilton et al., Nucl. Phys. B 60, 443 (1973)] are related to the standard function. The potential interest of $\Delta_\ell$ for the study of low-energy cross sections and weakly bound states is discussed in the framework of the proton-proton ${}^1S_0$ collision. The resonant state of the proton-proton collision is successfully computed from the extrapolation of $\Delta_\ell$ instead of the standard function. It is shown that interpolating $\Delta_\ell$ can lead to useful extrapolation to negative energies, provided scattering data are known below one nuclear Rydberg energy (12.5 keV for the proton-proton system). This property is due to the connection between $\Delta_\ell$ and the effective-range function by Hamilton et al. that is discussed in detail. Nevertheless, such extrapolations to negative energies should be used with caution because $\Delta_\ell$ is not analytic at zero energy. The expected analytic properties of the main functions are verified in the complex energy plane by graphical color-based representations.

The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants

We derive a closed formula for the determinant of the Hankel matrix whose entries are given by sums of negative powers of the zeros of the regular Coulomb wave function. This new identity applied together with results of Grommer and Chebotarev allows us to prove a Hurwitz-type theorem about the zeros of the regular Coulomb wave function. As a particular case, we obtain a new proof of the classical Hurwitz's theorem from the theory of Bessel functions that is based on algebraic arguments. In addition, several Hankel determinants with entries given by the Rayleigh function and Bernoulli numbers are also evaluated.

Triple differential cross sections for the ionization of formic acid by electron impact

Theoretical triple differential cross sections for the simple ionization of the two outermost valence orbitals of formic acid monomer are given in this work. The first Born approximation is used to describe the collision dynamics with one Coulomb wave function for the ejected electron. Single-center molecular wave functions are generated using the Gaussian 09 program. The cross sections are calculated for an average molecular orientation with the proper average method. The results show that the proposed framework provides better agreement with experimental data than the previously used orientation averaged molecular orbital approximation. Consequently, this work shows the importance of performing the orientation averaging on the cross sections and not on the molecular wave functions. It also provides a methodology that reduces the complexity of calculations while maintaining a good quality of the results.

Potential for α -induced nuclear scattering, reaction and decay, and a resonance-pole-decay model with exact explicit analytical solutions

The decay of $\alpha$ particle from a nucleus is viewed as a quantum resonance state of a two-body scattering process of the $\alpha$+daughter nucleus pair governed by a novel nucleus-nucleus potential in squared Woods-Saxon form. By the application of the rigorous optical model (OM) potential scattering (S-matrix) theory the genuineness of the potential for the system is established by giving good explanation of the elastic scattering and reaction cross sections data of the $\alpha$+nucleus pair. From the pole position in the complex momentum (k) plane of the S-matrix defined above, the energy and width of the resonance state akin to the decaying state of emission of $\alpha$ particle are extracted and from this width, the result of $\alpha$-decay half-life is derived to account for the experimental result of half-life in the cases of large number of $\alpha$-emitters including heavy and super-heavy nuclei. The S-matrix of the full OM calculation above is replaced by an analytical function expressed in terms of exact Schr\"{o}dinger solutions of a global potential that closely represents the Coulomb-nuclear interaction in the interior and the pure Coulomb wave functions outside, and the resonant poles of this S-matrix in the complex momentum plane are used to give satisfactory results of decay half-lives of $\alpha$ coming out from varieties of nuclei.

Orientation effect on the secondary-electron distributions for the HCl double ionization

Multiple differential cross sections of the double ionization of hydrogen chloride molecule impacted by electrons are here provided within the first Born approximation. The incident and scattered electrons are represented by plane wave functions, whereas the target is described by means of a single-center molecular wave function. The two ejected electrons are modeled by Coulomb wave functions and a Gamow factor is introduced to take into account the Coulomb repulsion between the two outgoing particles. In this work, we study the relative contributions of the four outermost orbitals of the HCl molecule to the double ionization process in the particular case of two electrons ejected from different subshells. The multiple differential cross sections are analyzed for specific target orientations and compared in terms of magnitude and shape. We report a strong dependence of the cross sections versus the target orientation by pointing out the signature of the well-known double ionizing processes, namely, the shake-off and the two-step 1 mechanisms.

Double differential cross sections for ion impact ionization of ammonia(NH3) and methane (CH4) molecules

SynopsisWe present a theoretical approach to calculate double-differential cross sections (DDCS) for single ionization of ammonia (NH3) and methane (CH4) molecules by the impact of proton at energies 0.25 MeV and 2 MeV respectively. The calculation are performed by the three-body formalism of three-Coulomb wave model (3C-3B). In the present formalism, the interaction of the projectile ion with the target is described by the Coulomb continuum wave functions in the initial channel. In the final channel, we have expressed the total wavefunctions as the product of pair-wise Coulomb wavefunctions among the ejected electron, projectile ion and the residual target ion, respectively.

Variational study of atomic hydrogen excitation by proton impact at intermediate velocities: target continuum states contribution

The Schwinger variational principle is used to evaluate the cross section for the excitation and Balmer-α emission of the hydrogen atom. The target is projected by proton impact at energies between 2 and 200 keV. To describe the strong coupling between the excitation and capture channels at intermediate impact energies, the Coulomb wave function, that is, exact continuum states of the hydrogen atom, have been added in the expansion of the scattering wave function. From comparison with the available experimental data and other theoretical findings we remark that taking into account the continuum states of the target is a good choice to describe the capture states at medium and low velocities, thus making the variational approach given in this work an easy way to successfully study the excitation of atoms by ion impacts at intermediate velocities.

Model of a fluxtube with a twisted magnetic field in the stratified solar atmosphere

We build a single vertical straight magnetic fluxtube spanning the solar photosphere and the transition region which does not expand with height. We assume that the fluxtube containing twisted magnetic fields is in magnetohydrostatic equilibrium within a realistic stratified atmosphere subject to solar gravity. Incorporating specific forms of current density and gas pressure in the Grad–Shafranov equation, we solve the magnetic flux function, and find it to be separable with a Coulomb wave function in radial direction while the vertical part of the solution decreases exponentially. We employ improved fluxtube boundary conditions and take a realistic ambient external pressure for the photosphere to transition region, to derive a family of solutions for reasonable values of the fluxtube radius and magnetic field strength at the base of the axis that are the free parameters in our model. We find that our model estimates are consistent with the magnetic field strength and the radii of Magnetic bright points (MBPs) as estimated from observations. We also derive thermodynamic quantities inside the fluxtube.